Infer Innovations from Model Containing Regression Component

Estimate a VAR(4) model of the consumer price index (CPI), the unemployment rate, and the gross domestic product (GDP). Include a linear regression component containing the current quarter and the last four quarters of government consumption expenditures and investment (GCE). Infer model innovations.

The residuals corresponding to the CPI and GDP growth rates exhibit heteroscedasticity because the CPI series appears to cycle through periods of higher and lower variance. Also, the first half of the GDP series seems to have higher variance than the latter half.

Input Arguments

Mdl — VAR modelvarm model object

VAR model, specified as a varm model object created by varm or estimate. Mdl must be
fully specified.

Y — Response datanumeric matrix | numeric array

Response data, specified as a numobs-by-numseries
numeric matrix or a
numobs-by-numseries-by-numpaths
numeric array.

numobs is the sample size. numseries is the
number of response series (Mdl.NumSeries).
numpaths is the number of response paths.

Rows correspond to observations, and the last row contains the latest observation.
Y represents the continuation of the presample response series in
Y0.

Columns must correspond to the response variable names in
Mdl.SeriesNames.

Pages correspond to separate, independent numseries-dimensional
paths. Among all pages, responses in a particular row occur at the same time.

Data Types: double

Name-Value Pair Arguments

Specify optional
comma-separated pairs of Name,Value arguments. Name is
the argument name and Value is the corresponding value.
Name must appear inside quotes. You can specify several name and value
pair arguments in any order as
Name1,Value1,...,NameN,ValueN.

Example: 'Y0',Y0,'X',X uses the matrix Y0 as presample responses and the matrix X as predictor data in the regression component.

'Y0' — Presample responsesnumeric matrix | numeric array

Presample responses providing initial values for the model, specified as the comma-separated
pair consisting of 'Y0' and a
numpreobs-by-numseries numeric matrix or a
numpreobs-by-numseries-by-numprepaths
numeric array.

numpreobs is the number of presample observations.
numprepaths is the number of presample response paths.

Rows correspond to presample observations, and the last row contains the latest
presample observation. Y0 must have at least Mdl.P
rows. If you supply more rows than necessary, infer uses only
the latest Mdl.P observations.

'X' — Predictor datanumeric matrix

Predictor data for the regression component in the model, specified as the comma-separated
pair consisting of 'X' and a numeric matrix containing
numpreds columns.

numpreds is the number of predictor variables
(size(Mdl.Beta,2)).

Rows correspond to observations, and the last row contains the latest observation.
infer does not use the regression component in the
presample period. Therefore, X must have at least as many
observations as are used after the presample period.

Note

If Y is a 3-D array, then infer horizontally concatenates the pages of Y to form a numobs-by-(numpaths*numseries + numpreds) matrix.

If a regression component is present, then infer horizontally concatenates X to Y to form a numobs-by-numpaths*numseries + 1 matrix. infer assumes that the last rows of each series occur at the same time.

infer removes any row that contains at least one NaN from the concatenated data.

infer applies steps 1 and 3 to the presample paths in Y0.

This process ensures that the inferred output innovations of each path are the same size and are based on the same observation times. In the case of missing observations, the results obtained from multiple paths of Y can differ from the results obtained from each path individually.

Loglikelihood objective function value associated with the VAR model Mdl, returned as a numeric scalar or a numpaths-element numeric vector. logL(j) corresponds to the response path in Y(:,:,j).

Algorithms

infer infers innovations by evaluating the VAR model Mdl with respect to the innovations using the supplied data Y, Y0, and X. The inferred innovations are

ε^t=Φ^(L)yt−c^−β^xt−δ^t.

infer uses this process to determine the time origin
t0 of models that include linear time trends.

If you do not specify Y0, then
t0 = 0.

Otherwise, infer sets
t0 to
size(Y0,1) – Mdl.P. Therefore, the
times in the trend component are t =
t0 + 1,
t0 + 2,...,
t0 + numobs,
where numobs is the effective sample size
(size(Y,1) after infer removes
missing values). This convention is consistent with the default behavior of
model estimation in which estimate removes the first
Mdl.P responses, reducing the effective sample size.
Although infer explicitly uses the first
Mdl.P presample responses in Y0 to
initialize the model, the total number of observations in Y0
and Y (excluding missing values) determines
t0.

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